#6699761

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1408 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,4,6)(3,7,8,5)(9,10)(11,14)(12,13)(15,16)(17,18,23,29)(19,30,34,25)(20,26,33,43)(21,38,24,27)(22,40,31,28)(32,42,41,44)(35,48,39,45)(36,47,37,46);(1,3)(2,5)(4,8)(6,7)(17,19,23,34)(18,25,29,30)(20,35,33,39)(21,32,24,41)(22,37,31,36)(26,45,43,48)(27,42,38,44)(28,46,40,47);(2,6)(5,7)(10,13)(14,16)(17,20,23,33)(18,26,29,43)(19,31,34,22)(21,37,24,36)(25,40,30,28)(27,47,38,46)(32,39,41,35)(42,45,44,48);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,21,23,24)(18,27,29,38)(19,32,34,41)(20,36,33,37)(22,39,31,35)(25,42,30,44)(26,46,43,47)(28,45,40,48);(3,8)(5,7)(17,22,23,31)(18,28,29,40)(19,33,34,20)(21,39,24,35)(25,43,30,26)(27,45,38,48)(32,37,41,36)(42,47,44,46)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=fg,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6606957);